16-Civ-B4 Engineering Hydrology · December 2017

Question 3 of 7: Point and Areal Precipitation, and Stream-Flow Measurement

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Question 3: Point and Areal Precipitation, and Stream-Flow Measurement (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

3(i) — The Thiessen-polygon technique, and the effect of a sparse gauge network (6 marks)

A rain gauge measures precipitation at a point, but a water balance needs the average depth over an area. The Thiessen technique supplies that average by assuming that each gauge is representative of the ground that lies closer to it than to any other gauge. Construction is purely geometric: plot the gauges, join every neighbouring pair with a straight line, erect the perpendicular bisector of each line, and let the bisectors intersect. The polygon enclosing each gauge is the region nearer to it than to any other, and its area within the watershed boundary becomes that gauge’s weight. Gauges outside the watershed are included in the construction whenever their polygons overlap it, which is one of the method’s useful features.

Given. The storm-total catch and Thiessen polygon area at each of four gauges on a 190 km² watershed.

Find. The areal-mean precipitation and the total storm volume, and the error the arithmetic mean would introduce.

Approach. Form the area-weighted mean, convert it to a volume, and compare with the unweighted mean to expose the weighting effect.

1. Form the weighted sum. $$\sum A_i P_i = (42)(78) + (65)(52) + (31)(96) + (52)(34) = 3276 + 3380 + 2976 + 1768 = 11\,400\ \text{km}^2\!\cdot\!\text{mm}$$
2. Divide by the total area. $$\bar{P} = \frac{\sum A_i P_i}{\sum A_i} = \frac{11\,400}{190} = \boxed{60.0\ \text{mm}}$$
3. Convert to a storm volume. $$V = \bar{P}A = (0.0600)(190 \times 10^6) = 11.4 \times 10^6\ \text{m}^3$$
4. Contrast with the arithmetic mean. $$\bar{P}_{\text{arith}} = \frac{78+52+96+34}{4} = 65.0\ \text{mm}, \qquad \text{an overestimate of } \frac{65.0-60.0}{60.0} = 8.3\%$$ The unweighted mean is high because the wettest gauge, G3, happens to sit in the smallest polygon; weighting demotes it to its proper share.

The effect of a limited number of gauges. The method’s accuracy depends entirely on whether the nearest-gauge assumption is true, and a sparse network attacks that assumption in three ways. First, the polygons become large, so each gauge is asked to represent terrain that may differ from it in elevation, aspect and exposure — a serious matter in mountainous British Columbia, where orographic gradients can double the depth over a few kilometres, and where the polygon geometry knows nothing about topography. Second, the whole spatial structure of the storm may be missed: a convective cell smaller than a polygon either lands on a gauge, and is then extrapolated over the whole polygon, or falls between gauges and is not recorded at all, and the two errors are of opposite sign and comparable magnitude. Third, the weights themselves become unstable, so that adding or losing a single gauge redraws the polygons and shifts the answer materially. Taking the same watershed with only two of the four gauges surviving and the polygons redrawn to 88 and 102 km²,

which is 6.7 % high on a single storm, and would be arbitrarily high or low on the next one. The practical remedies are to supplement the gauges with weather radar or gridded reanalysis so that the storm structure is observed rather than assumed; to use the isohyetal method where topographic control is strong, since it lets the analyst impose orographic knowledge that Thiessen cannot; and to report the areal mean with an uncertainty band rather than as a single number, which matters because that uncertainty propagates directly into every runoff volume and every calibrated model parameter downstream.

3(ii) — Stage and rating curves for measuring storm flow (6 marks)

Discharge cannot be measured continuously; water level can. The stage–discharge rating curve is the device that converts one into the other, and it is the basis of essentially every published streamflow record, including the whole Water Survey of Canada HYDAT archive.

The procedure has three parts. Establishing the rating requires a set of direct discharge measurements — by current meter on a wading or cableway section, or now more usually by acoustic Doppler current profiler from a boat or tethered float — each paired with the stage at the moment of measurement. The pairs are fitted with a power law of the form $Q = a(H - H_0)^b$, in which $H_0$ is the stage of zero flow, so that the fit reflects the physics of a control section rather than being an arbitrary curve. Recording the stage is continuous, by float-and-counterweight in a stilling well, by pressure transducer, or by a non-contact radar sensor, logged at fifteen-minute intervals or finer through a storm. Applying the rating converts every stage reading to a discharge, and integrating the resulting hydrograph over the event gives the storm runoff volume.

Given. A gauging station whose rating is $Q = 42.5(H - 0.35)^{2.15}$ with $Q$ in m³/s and $H$ in metres, and a recorded storm peak stage of $H = 3.60$ m. The stage sensor is accurate to $\pm 0.05$ m.

Find. The peak discharge and the discharge uncertainty that the stage uncertainty implies.

1. Apply the rating at the peak stage. $$Q = 42.5(3.60 - 0.35)^{2.15} = 42.5(3.25)^{2.15} = 42.5(12.605) = \boxed{535.7\ \text{m}^3/\text{s}}$$
2. Propagate the stage error. Differentiating the power law logarithmically, $$\frac{\Delta Q}{Q} = b\,\frac{\Delta H}{H - H_0} = 2.15\,\frac{0.05}{3.25} = 0.033$$ so the discharge is uncertain by 3.3 %, or $\pm 17.7$ m³/s.

Note the amplification: the exponent $b$ multiplies the relative stage error, so a rating with a steep exponent converts a small level error into a considerably larger flow error. That is before any error in the rating itself, which is usually the larger term at flood stage.

A technique that improves the accuracy of the measured flow. The single most effective improvement is to install an index-velocity station: a side-looking acoustic Doppler velocity meter mounted in the channel measures the mean velocity of a sampled path continuously, and discharge is then computed as the product of that velocity, a rating of mean-section velocity against index velocity, and the stage–area relation. This removes the method’s worst weakness, which is that a single-valued stage–discharge rating cannot represent unsteady flow. During a flood wave the water-surface slope is steeper on the rising limb than on the falling limb, so the same stage carries more discharge on the way up than on the way down; the true relation is a loop, and the single-valued rating splits the difference, under-reading the rising limb and over-reading the recession. An index-velocity station measures through the loop rather than assuming it away. Where that is not practicable, the next-best measures are to obtain direct ADCP gaugings at high flow so that the upper end of the rating is measured rather than extrapolated, to survey the control section after every major event since scour or deposition shifts the rating and requires a new one, and to apply the Jones formula as an unsteady-flow correction to the rated discharge.

3(iii) — Comparing a natural and a human-induced mechanism of streamflow change (8 marks)

Two mechanisms are compared: a natural mechanism, the transition of a basin’s snowmelt regime under a warming climate, and a human-induced mechanism, urbanisation of the basin’s lower catchment. Both change the flow, but they change different parts of it, over different timescales, and they call for different engineering responses.

Mechanism and timescale. The snowmelt mechanism operates through storage: a snowpack is a reservoir that accumulates winter precipitation and releases it over a few weeks in spring. Warming reduces the fraction of precipitation stored as snow and advances the release, so the annual hydrograph shifts rather than simply growing. The change accumulates over decades and is detected statistically, as a trend in freshet timing and in the ratio of spring to annual flow, rather than observed in any single year. Urbanisation operates through the loss of infiltration and the acceleration of conveyance: impervious roofs and pavements convert rainfall to runoff essentially in full, and storm sewers deliver it to the channel far faster than overland flow across a pervious basin would. The change is abrupt, arriving with the subdivision, and it is confined to the developed area.

Effect on the flow regime. The snowmelt change moves water between seasons without necessarily changing the annual volume: the spring peak arrives earlier and is usually somewhat lower, while winter flows rise because more precipitation runs off rather than being stored, and late-summer flows fall because the pack is exhausted sooner. Rain-on-snow events become more frequent at mid elevations, and these produce some of the most severe floods on record in coastal British Columbia. Urbanisation raises the runoff coefficient several-fold and shortens the time of concentration, so the flood peak from a given storm may increase by a factor of two to five while the time to peak halves. The frequent, small events change most of all: a storm that formerly produced no measurable runoff now produces a full hydrograph, so the effective bankfull frequency rises and the channel enlarges by erosion. Base flow generally falls, because the recharge that formerly sustained it has been diverted to the storm sewer.

Spatial extent and reversibility. The snowmelt mechanism is regional, affects gauged and ungauged basins alike, and is not reversible on any engineering timescale; the appropriate response is adaptive — re-analysing the flood-frequency record for non-stationarity, revising reservoir rule curves so that flood-control drawdown occurs earlier, and designing for a shifted rather than a historical distribution. The urbanisation mechanism is local, is caused by an identifiable decision, and is at least partly reversible or preventable through design: stormwater detention, infiltration practices such as bioswales and permeable pavement, and imperviousness limits in the zoning bylaw can hold the post-development peak to the pre-development value, which is precisely the standard applied in Problem 7.

What they have in common, and why it matters for design. Both mechanisms violate the stationarity assumption on which conventional flood-frequency analysis rests. A gauge record that spans a period of warming, or a period during which its basin urbanised, is not a sample from one distribution, and fitting a Gumbel or log-Pearson curve to it produces a design flood that describes neither the past nor the future. The correct treatment is the same in both cases: identify the change point or trend, either adjust the record to a common basin condition or fit a non-stationary distribution, and state the basin condition to which the design flood applies. The difference is that urbanisation can be quantified from mapped imperviousness and modelled directly, whereas the climate signal must be handled through scenarios and carried as an explicit uncertainty.